Equation 18 · How Fast Can a Horizon Learn Which Path You Took?

What does this equation mean?

T_\chi(\delta)= \inf\{T:\chi_H(T)\geq1-\delta\},

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Inputs and operationsinfT:chi_H(T) ≥ 1-delta

Result or conditionT_chi(delta)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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T_\chi

Symbol T_chi

$T_c$ hi is part of the quantity the equation computes from the expression on the right.

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\delta

Symbol delta

the define a confidence deficit.

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T

Symbol T

T is part of the quantity the equation computes from the expression on the right.

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\chi_H

Symbol chi_H

ch $i_H$ is one of the signed contributions combined to compute the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

where $\Gamma$ has units of inverse time in the explicitly named time coordinate. Define a confidence deficit $\delta$ . A Holevo-ceiling time is $T_\chi(\delta)= \inf\{T:\chi_H(T)\geq1-\delta\}$ . and a minimum-error decision time is

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